Asymptotics of Analytic Integrals and the Beurling-Malliavin Theory

本提案的统一主题是研究希尔伯特变换及其最接近的数学亲戚，柯西变换和里兹变换，在各种应用中出现的非齐次设置。这个项目中包含的应用在大约50年前就受到了分析人士的关注——完备性和极小性问题，特别是Beurling-Malliavin理论、Beurling-Levinson型的“间隙和密度”定理、Cartwright和paly - wiener空间理论等。这些领域包含线性复分析的一些最深刻的结果。即使是现代的论证也需要数百个寻呼机，而且有些证明（比如伯灵-马利亚文乘数定理的证明）看起来仍然完全神秘。光谱分析的需要要求用一种新的方法来解决这些问题，并对经典结果进行扩展。现代复杂分析的强大技术应该会对经典理论进行广泛的推广，并对光谱问题进行有效的应用。本项目主要研究复杂分析及其应用。复杂分析是数学的经典领域，在纯粹研究和应用研究中继续发挥着重要作用。复分析的典型对象之一就是所谓的希尔伯特变换。希尔伯特变换的研究使人们能够理解复可微函数在其定义域边界附近的行为。尽管希尔伯特变换是所有数学中研究最多的对象之一，但人们还远远没有完全理解它。此外，应用领域的新发展，如固态物理和微分方程的数学模型，需要对经典结果进行相当大的扩展。这个项目的目标是提供这样的扩展，并在分析和数学物理的几个领域应用新的结果。这些应用包括弦方程和薛定谔方程的谱问题，薛定谔方程描述了量子力学中的波传播。

The unifying theme of this proposal is the study of the Hilbert transform and its closest mathematical relatives, the Cauchy transform and the Riesz transform, in non-homogeneous settings appearing in various applications. The applications included in this project were in the center of attention of analysts about 50 years ago -- the completeness and minimality problems, specifically the Beurling-Malliavin theory, the "gap and density" theorems of Beurling-Levinson type, the theory of Cartwright and Paley-Wiener spaces, etc. These areas contain some of the deepest results of linear complex analysis. Even modern expositions require hundreds of pagers with some proofs (like the proof of the Beurling-Malliavin multiplier theorem) still looking completely mysterious. The needs of spectral analysis call for a new approach to these problems and for an extension of the classical results. Powerful techniques of modern complex analysis should give rise to vast generalizations of the classical theory and effective applicationsto spectral problems.This project focuses on complex analysis and its applications. Complex analysis is aclassical area of mathematics that continues to play an important role inboth pure and applied studies. One of the canonical objects of complex analysisis the so-called Hilbert transform. Studies of the Hilbert transform allow one to understand the behavior of complex differentiable functions near the boundary of their domains. Despite being one of the most studied objects in all of mathematics,Hilbert transform is far from being completely understood. Moreover, new developmentsin applications, such as mathematical models of solid state physics and differential equations, require considerable extensions of classical results. The goal of this project is to provide such extensions and to apply new results in several areas of analysis and mathematical physics. Among such applications are spectral problems for the string equation and the Schroedinger equation, which describes wave propagation in quantum mechanics.